GBTUL 1.0 Buckling and Vibration Analysis of Thin-Walled Members

USER MANUAL

Rui Bebiano Nuno Silvestre Dinar Camotim

Department of Civil Engineering and Architecture, DECivil/IST

Technical University of Lisbon – Portugal

2010

1. Introduction

The code GBTUL 1.0, which implements recently developed formulations

of the Generalized Beam Theory (GBT), performs linear buckling

(bifurcation) and vibration analyses of elastic thin-walled members.

The objective of the Program Tutorial is to provide concise descriptions of

all the commands, data entries and results outputs available at the GBTUL

graphical user interface.

GBTUL 1.0 (acronym for “GBT at the Technical University of Lisbon”) is a

freeware code, meant to provide the users with a graphical and easy-to-use

structural analysis tool. Being based on GBT, it allows the users to benefit

from the technique’s unique modal features. For more information,

including access to the electronic forms of the manuals referenced above,

visit the program website http://www.civil.ist.utl.pt/gbt.

2. Program Scope and Structure

The code GBTUL (more specifically, its 1.0 version) performs buckling

(bifurcation) and vibration analyses of elastic thin-walled members with

arbitrary open cross-sections (i.e., cross-sections containing closed cells cannot

be handled) – see Figure 2.1. Moreover, each wall can be made of one or

more isotropic or specially orthotropic materials.

Concerning the member end support conditions, the code covers (i) pinned

(simply supported, S-S), (ii) cantilever (C-F), (iii) fixed (C-C) and (iv) fixed-

pinned (C-S) members – moreover, it is possible to specify different support

conditions for the various deformation modes (e.g., for bending and torsion).

As for the (pre-buckling) loadings, they may stem from combinations of (i) end

moments or axial forces, (ii) distributed, and (iii) concentrated forces (the

transverse loads acting upon the shear center). However, no effect of applied

loading (i.e., reduction of stiffness/frequency due to geometrically non-

linear work of the acting stresses) is accounted for in vibration analyses.

Moreover, two types of member analysis are available: (i) the Analytical

Solution, only for simply supported (S-S) members under uniform loads,

and (ii) the Numerical Solution, always applicable and involving a

discretisation of the member length into GBT-Based finite elements (see

Table 2.1). Whenever possible, the Analytical Solution should be used,

because of the lower computational time required and its simpler inputs

(with relation to an equivalent Numerical Solution).

(a) (b)

Figure 2.1: Examples of member cross-sections: (a) open (handled by GBTUL 1.0) and (b)

closed (not handled by GBTUL 1.0)

Table 2.1: Differences between Analytical and Numerical Solutions

Type of

Solution

Support

Conditions

Pre-Buckling

Loads

Eigensystem

Solver

Number of

Halfwaves

Analytical S-S1 Uniform Cholesky Fact. User-provided

4

Numerical

S-S2

Uniform or

Non-Uniform3

Cholesky Fact. or

Stodola Method

Automatically

computed

C-F2

C-C2

C-S2

Notes: 1 The same (S-S) for all deformation modes.

2 To be provided independently for (i) major and (ii) minor axis bending, (iii)

torsional and distortional modes and (iv) local-plate modes. 3 Resulting from (i) unequal end moments and (ii) distributed or (iii) point loads.

4 A set of numbers from 1 to the maximum value, specified by the user.

The user is able to provide an arbitrarily long list of member lengths (L

values), so that the code produces a curve describing the evolution of b

(buckling load parameter) or (natural angular frequency) with L as well as

the corresponding Pi vs L modal participation diagrams. The buckling or

vibration modes are represented by means of either (i) 3D deformed

configurations of the entire member, including interactive visualisation tools, or

(ii) 2D deformed configurations of any given cross-section. Furthermore, the

code output data is also saved in formatted text files, which makes it very easy

to process them by means of other spreadsheet applications (e.g., Microsoft

Excel).

The GBTUL 1.0 graphic interface involves the sequence of four screens shown

in Figure 2.2: while the first three deal with data input, the fourth one provides

the result output. This sequence is closely related to the performance of a GBT

analysis (see GBT Theoretical Reference): (i) while Screens 1, 2 and 3 concern

the inputs associated with the cross-section analysis, deformation mode

selection and member analysis, (ii) Screen 4 displays the sought buckling or

vibration solution.

Screen 2

Deformation mode

display and

selection

Screen 1

Type of analysis Member material

Section geometry

Screen 4

b-L, -L, Pi-L curves

2D, 3D configurations

Screen 3

Lengths Applied Loads

Support Conditions

Inputs Outputs

Figure 2.2: GBTUL – structure of the graphic interface

3. Input Data: Screens 1-3

3.1 Screen 1: Cross-section Analysis

The first screen of GBTUL 1.0 prompts the user for the data concerning (i)

the material, (ii) the cross-section geometry and nodal discretisation and

(iii) the type of analysis pretended (i.e., buckling or vibration). The

interface includes templates corresponding to several usual cross-section

geometries (e.g., C, Z, I sections), which the user should use whenever

possible, since the “Natural Nodes” and “Section Walls” tables are

intended for general-type cross-sections.

Figures 3.2-3.3 present, respectively, a general overview of the Screen 1

and the dialogue boxes related – in both cases, all the interface objects are

identified and the corresponding usage is explained next.

Notes concerning the features of Screen 1:

(1) Screen tabs: The first four tabs correspond to each of the four

screens involved in the procedure of a GBTUL analysis. The tabs are

intended for the user to review or change input data provided before,

and not to proceed to the next screen – rather, this should be done by

clicking button NEXT. As for the fifth and last screen, it displays

some information about the authors of the program.

(2) Material model: Allows the user to specify the orthotropic material

properties: (i) longitudinal (Exx) and transversal (Ess) elastic

modulli, (ii) Poisson’s ratios (Uxs and Usx), (iii) distortion modulus

(Gxs) and (iv) volumetric mass density (ro). Different materials can

be specified in further lines, each one should be given a different

reference number (#Mat). For isotropic members (e.g., steel beams)

it is easier to use the “Isotropic” template (see (3)). For buckling

analyses, a unit value may be assigned to ro.

(3) Isotropic material template: Prompts the user for the mechanical

properties defining the isotropic material: (i) elastic modulus (E), (ii)

Poisson’s ratio (u) and (iii) volumetric mass density (ro). For

buckling analyses, a unit value may be assigned to ro.

(4) Natural nodes: In this table, the user introduces the “X” (horizontal)

and “Y” (vertical) coordinates of the natural nodes, i.e., the points

defining the ends and intersections of the plates that compose the

member cross-section. For an N-plated cross-section, the number of

natural nodes is always equal to N+1, and each one should be defined

by its own reference number (“#Nodes”). The nodes should be

numbered successively and according to their order of placement in

the cross-section (see Figure 3.1) – for branched1 sections, see GBT

Theoretical Reference, Part 3.

(5) Section walls: In this table the user enters the data defining each of

the cross-section walls, namely:

#Wall: The wall reference number;

Node I: The reference number of the initial node of the wall;

Node J: The reference number of the end node of the wall;

Order: The “order” of the wall – for branched sections1 see

GBT Theoretical Reference, Part 3. For unbranched1 sections,

insert “0” for all the walls;

#Mat: The reference number of the wall material;

Inodes: The number of intermediate nodes of the

discretization;

Tick: The wall (uniform) thickness.

The walls, as well as the natural nodes, should be defined in a

consistent and manner, i.e., numbered according to their placement

in the cross-section geometry, as Figure 3.1 shows.

(6) C/U-sections template: Define the geometry of the lipped channel,

by entering the web (bw), flanges (bf) and lips (bl) widths and

numbers of intermediate nodes (INodes), the lip angle (º), and the

thickness (t). For plain channels (U-sections), just enter bl=0. For a

Hat-section, enter º=-90.

1 A cross-section is said to be unbranched if no more than 2 walls share any of its nodes (e.g., C, Z, U-

sections), and branched otherwise (e.g., I, T-sections).

Figure 3.1: Two examples of natural nodes and walls numbering: - natural nodes,

- walls.

(7) Rack sections template: Define the geometry of the “Rack”

sections, by entering the web (bw), flanges (bf), inner lips (bl1)

and outer lips (bl2) widths and numbers of intermediate nodes

(INodes), the inner and outer lips angles (1º and 2º), and the

thickness (t). For a “Return lips” section, enter 2º=180.

(8) Z-sections template: Define the geometry of the Z-section, by

entering the web (bw), flanges (bf) and lips (bl) widths and

numbers of intermediate nodes (INodes), the lip angle (º), and the

thickness (t). For plain Z-sections, just enter bl=0.

(9) I/T-sections template: Define the geometry of the I-section, by

entering the web (bw) and flanges (bf1 (top) and bf2 (bottom))

widths, numbers of intermediate nodes (INodes) and thicknesses

(tw, tf1 and tf2). For T-sections, just enter bf2=0.

(10) Angle sections template: Define the geometry of the angle section,

by entering the web (bw) and flange (bf) widths and numbers of

intermediate nodes (INodes), and the thickness (t).

(11) Plate section template: Define the geometry of the plate section, by

entering its height (h), the thickness (t) as well as the number of

intermediate nodes (INodes). Consider at least 2 intermediate

nodes in the plate. Besides, note that a minimum slenderness of h/t=5

is expected, at least for an accurate analysis of local deformation.

(12) Section graphic representation window: Plots the cross-section, as

well as some related items (see 13), for the user to visualize/confirm

the data provided.

(13) Visualization tools: Check the checkboxes for the following items

to be displayed: Natural Nodes (i.e., the #Nodes numbers),

Intermediate nodes, Coordinate System x-y (the one associated with

the natural nodes), Wall Numbers (i.e., the #Walls numbers), Wall

Orders and Materials (i.e., the #Mat numbers).

(14) Analysis Type: Select the type of analysis desired: Stability

(i.e., linear buckling) or Vibration (i.e., free vibration).

(15) NEXT: Move on to the Screen 2, to proceed to the GBT Mode

Selection inputs screen.

Screen 1 – Cross-Section Analysis

Figure 3.2: GBTUL 1.0 – Overview of Screen 1

Screen tabs

- see (1)

Isotropic material

template - see (3)

and Fig. 3.3(a)

Clears all the

table cells

Material Model

input – see (2)

Natural Nodes

input – see (4)

Section Walls

input – see (5)

Cross-Section

Templates – see

(6)-(11) and

Figs. 3.3(b)-(f)

Section graphic

representation

window

– see (12)

Analysis Type

– see (14)

Button NEXT

– see (15)

Visualisation

tools – see (13)

Screen 1 – Cross-Section Analysis (dialogue boxes)

(a) Isotropic template – see (3)

(b) C/U-sections template – see (6)

(c) Rack section template – see (7)

(d) Z-section template – see (8)

(e) I/T-section template – see (9)

(f ) Angle section template – see (10)

Figure 3.3: GBTUL 1.0 – Dialogue boxes of Screen 1: (a) Isotropic material template and (b)-(f) cross-section geometry templates.

3.2 Screen 2: GBT Mode Selection

The second screen of GBTUL 1.0 shows the results of the cross-section

analysis, i.e., (i) the most relevant geometrical properties (e.g., area, inertia

moments, etc.), (ii) the stiffness matrices (linear and geometrical) and (iii)

the deformed configurations of the GBT deformation modes. Moreover,

this screen allows the user to select the set of GBT deformation modes to

be included in the analysis. Figure 3.4 shows a general overview of Screen

2 where the objects are identified and explained in the following notes.

Notes concerning the features of Screen 2:

(16) Cross-section geometrical properties: A list of eleven cross-section

geometrical properties (only available for isotropic members): Area

(A), major and minor inertia moments (I1, I2), warping (G) and

torsional (J) constants, center of mass (x.cg, y.cg) and shear

center (x.sc, y.sc) coordinates and asymmetry factors (b1, b2).

(17) GBT stiffness matrices: By pressing the buttons, the GBT stiffness

matrices are displayed in separate dialogue boxes (see the GBT

Theoretical Reference, Part 2, for physical meaning of the matrices).

(18) Mode graphic representation window: Plots the in-plane or out-of-

plane displacement fields associated with the selected GBT

deformation mode (see (19)). The undeformed cross-section is

represented in yellow, while the modal configuration does in red.

(19) Mode visualization tools: The mode representation window (see

(18)) displays the configuration of the mode which number appears

on the choice box – by using the “<” and “>” buttons, the user may

choose the mode to be shown. Furthermore, the buttons “in-plane

displacements” and “warping displacements” allows

the choice between in-plane and out-of-plane displacements.

(20) Mode selection tools: By default, all the available deformation

modes (see the GBT Theoretical Reference, Part 4, for comments on

the number of GBT modes obtained) are to be considered. In order to

choose a subset of these modes, the user may write their number

directly on the text box, or use the button “Pick mode” button to

pick the mode currently displayed at the graphic window.

(21) NEXT: Move on to the Screen 3, to proceed to the Member Analysis

inputs screen.

Screen 2 – GBT Mode Selection

Figure 3.4: GBTUL 1.0 – Overview of Screen 2

Cross-section

geometrical pro-

perties - see (16)

and Fig. 3.2(a)

Mode

visualization

tools– see (19)

Mode graphic

representation

window

– see (18)

Button NEXT

– see (21)

Mode selection

tools – see (20)

GBT stiffness

Matrices - see (17)

3.3 Screen 3: Member Analysis

In the third screen of GBTUL the user (i) chooses the type of solution

(analytical or numerical – see the GBT Theoretical Reference, Part 5) and (ii)

specifies the member lengths, loading and support conditions. The specific

details associated with the two types of solution are introduced on the

corresponding tabs (“Analytical Solution” and “Numerical

Solution”), on the left side of the screen.

Figures 3.5 and 3.6(a)-(b) show, respectively, a general overview of the

Screen 3, the “Numerical Solution” tab and the “Log-Uniform”

dialogue box – in all cases, the interface objects are identified and

explained in the following notes.

Notes concerning the features of Screen 3:

(22) Analytical/Numerical tabs: These tabs prompt the user for the data

required to perform the Member Analysis (i.e., the resolution of the

member equilibrium equations) by the analytical or the numerical

procedure, respectively. The analytical solution is applicable only for

simply supported members, subjected (in the case of a buckling

analysis) to uniform loadings. The numerical solution, which

involves the longitudinal discretization of the member into GBT-

based beam finite elements (see 34), is always applicable. See Table

2.1 for a comparison between the features of these two procedures.

Only the tab associated to the procedure to be used need to be filled

by the user – it is important to check the checkbox at the bottom left

corner of the tab, to confirm the choice.

(23) Loading (Analytical Solution): Enter the load parameters defining

the pre-buckling, consisting of a combination of uniform axial force

(N – positive for compression), major (My – positive for compression

on the upper part of section) and minor (Mz – positive for

compressions on the left part of section) axis bending moment and

bimoment (B) – note that the bending moments act about the

principal axes of the cross-section, and not necessarily about x and y

axes.

(24) Number of halfwaves (Analytical Solution): Enter the maximum

number of longitudinal halfwaves to be exhibited by the (sinusoidal)

buckling or vibration mode – e.g., if this number is 3, the resulting

modes will exhibit between 1 and 3 halfwaves. For single-halfwave

buckling or fundamental vibration modes enter “1”.

(25) Number of intervals (Analytical Solution): This entry concerns

only the 2D and 3D graphical representations to be shown in Screen

4 – it defines the number of longitudinal intervals that define (i) the

number of cross-sections available for 2D representations and (ii) the

quality of the 3D representation. The default value, “10”, is fairly

enough for the 3D representation of single or two-halfwave modes,

but might be too low for modes exhibiting higher number of

halfwaves.

(26) Plot Member: Plots a representation of the member, i.e. its

supporting conditions and loading, for the conditions specified.

(27) Solution checkbox: Confirms the user’s choice between Analytical

or Numerical solution – it is necessary to check the due checkbox, in

addition to fill the tab and plot the member.

(28) Number of Eigenmodes: Defines the number of buckling or

vibration modes to be determined for each length. For the first

(critical) buckling or fundamental vibration mode, enter “1”.

(29) Member Graphic Representation Window: Represents graphically

the members to be analyzed, including the supporting conditions and

pre-buckling loading. For the Numerical Solution, the points

corresponding to end nodes of the finite elements are also presented.

The representation can refer to either x-y or x-z plans (see (30)).

(30) View plan: Allows the choice of the view plan to be represented on

the graphical window above (see (29)): (i) x-y corresponds to

longitudinal and major axis, and represents the support conditions

and loading pertaining to major axis bending, while (ii) x-z is the

same for minor axis bending. After changing the view plan, one must

press the Plot Member button for the view to be plotted.

(31) Log-uniform: This tool easily generates a list of lengths that are

equally spaced on a logarithmic scale, the one usually used in load

(or frequency) versus length curves. In the corresponding dialogue

box (see Fig. 3.6(b)), the entries (i1) “Number of lengths”, (ii1)

“From L =” and (iii1) “To L =” prompt, respectively, for (i2) the

number of lengths in the list, and (ii2) the lowest and (iii2) the highest

length values.

(32) Lengths: On this text box, the user writes the list of lengths of the

members to be analyzed – at least 2 values should be given. The

values should be written in ascending order. By default, a list of 63

lengths is provided. For a list of lengths equally spaced on a

logarithmic scale, use the “Log-Uniform” tool (see (31), and Fig.

3.6(b)).

(33) NEXT: Move on to perform the main analyses and then Screen 4,

with the output of the results.

(34) Number of GBT finite elements (Numerical Solution): Enter the

number of GBT-based beam finite elements of the (uniform)

longitudinal discretization. While the default number, “10” has

proved enough for accurate single to three-halfwave mode estimates,

a higher number should be provided if modes exhibiting higher

number of halfwaves are expected. For more information on this

finite element, see GBT Theoretical Reference, Part 5.

(35) Eigensystem Solver (Numerical Solution): The two mathematical

procedures to solve the GBT finite element eigensystem are (i) the

complete solution of the eigensystem by using the Cholesky

factorization, or (ii) the simplified solution, by the Stodola method

(faster but less accurate). The user is recommended to use the first

option, except for larger problems.

(36) Modal Boundary Conditions (Numerical Solution): By checking

the corresponding radio buttons, this tool allows the specification of

distinct support conditions to four types of GBT modes: mode 2

(major axis bending), mode 3 (minor axis bending), modes 4 and the

distortional ones (4-D) and the local-plate modes (LP). The four

types of support conditions available are: simply supported (S-S),

clamped-free (C-F – cantilever), clamped-clamped (C-C) and

clamped-supported (C-S). For more information on this modal

support conditions, see GBT Theoretical Reference, Part 5.

(37) Loading: axial force (Numerical Solution): Enter the parameters

defining the applied constant axial force (N), and distributed

longitudinal load (nx) – both positive for compression.

(38) Loading: major axis bending moment (Numerical Solution):

Enter the parameters defining the applied left (My1) and right (My2)

end moments, the transversal uniformly distributed load (py) and

one transversal point load (Qy), acting at the longitudinal coordinate

x=aL, where parameter a is also to be provided (“a” – e.g., for a

mid-span point load, enter “0.5”). The transversal loads act over the

shear centre axis.

(39) Loading: minor axis bending moment (Numerical Solution):

Enter the parameters defining the applied left (Mz1) and right (Mz2)

end moments, the transversal uniformly distributed load (pz) and

one transversal point load (Qz), acting at the longitudinal coordinate

x=aL, where parameter a is also to be provided (“a” – e.g., for a

mid-span point load, enter “0.5”). The transversal loads act over the

shear centre axis.

(40) Loading: bimoment (Numerical Solution): Enter the value of the

applied bimoment. Only uniform bimoments are dealt with by

GBTUL 1.0.

3.4 “DOS-like” interface window

After pressing button NEXT on Screen 3, the program starts to perform the

analyses for all the members (lengths). During this process, a DOS-like

interface appears, providing the user with some information about the

analyses being performed – in the case of the Numerical Solution (always

much more time-consuming that the Analytical Solution), this information

includes (i) the estimated time that the process will take and (ii) some

results already available (namely, the first buckling loads or vibration

frequencies of the members already analyzed).

Screen 3 – Member Analysis

Figure 3.5: GBTUL 1.0 – Overview of Screen 3

Analytical/Numeri-

cal tabs- see (22)

Number of intervals

– see (25)

Member graphic

representation

window

– see (29)

Button NEXT

– see (33)

Lengths – see (32)

Loading - see (23)

Number of

halfwaves - see (24)

Plot member

– see (26)

Solution checkbox

– see (27)

Number of eigen-

modes – see (28)

Log-uniform – see

(31) and Fig. 3.6(b)

View Plan

– see (30)

Screen 3 – Member Analysis (dialogue boxes)

(a) (b) – see (31)

Figure 3.6: GBTUL 1.0, Screen 3: (a) “Numerical Solution” tab and (b) “Log-Uniform” dialogue box.

Number of GBT

finite elements

– see (34)

Eigensystem

solver - see (35)

Modal boundary

conditions - see (36)

(i)

Solution checkbox

– see (27)

(i) Plot Member

– see (26)

Loading – see

(37)-(40) and

Figs. 3.7(a)-(d)

Screen 3 – Member Analysis (dialogue boxes)

(a) Loading: axial force – see (37)

(b) Loading: major axis bending moment – see (38)

(c) Loading: minor axis bending moment – see (39) (d) Loading: bimoment – see (40)

Figure 3.7: GBTUL 1.0, Screen 3: Dialogue boxes concerning the loading for the Numerical Solution

4. Output Data: Screens 4 and Text Files

The results of the analyses performed are represented graphically in Screen

4 as (i) plots of buckling or vibration curves, providing the variation of the

buckling load parameter or natural frequency with the member length L,

(ii) modal participation diagrams, and (iii) 2D or 3D representations of the

member buckling or vibration modes. In addition, all the data are also

recorded in formatted text files, making it possible any further processing.

4.1 Screen 4: Results

Screen 4, the last of the GBTUL 1.0 analysis procedure, presents graphical

outputs of the analyses, consisting of (i) buckling load or vibration

frequency vs. length curves, and (ii) 2D or 3D configurations and (iii)

modal participation diagrams of the buckling or vibration modes.

Figures 4.1, 4.2(a) and 4.2(b)-(c) depict, respectively, a general overview

of Screen 4, the Mode Selection dialogue box and the 2D and 3D

representation windows.

Notes concerning the features of Screen 4:

(41) Case Selection: Use the “<” and “>” buttons to select the member

length (top buttons) and buckling or vibration mode (bottom

buttons), to which refer (i) the summary of results and (ii) the 2D or

3D representations.

(42) Cross-Section: Use the “<” and “>” buttons to select the member

cross-section to which corresponds the 2D graphical representation.

The cross-section is identified by its x/L coordinate, and the number

of cross-sections available is equal to 1 plus the number of intervals

(for Analytical Solution) or the number of finite elements (for

Numerical Solution) defined at Screen 3.

(43) GBT modes: Use this tool to select a sub-set of deformation modes

– from within those selected at Screen 2 – to be included on the 2D

or 3D graphical representation. The numbers of the modes to be

considered can be entered directly on the dialogue box (see Fig.

4.2(a)). This enables, for instance, to isolate the contribution of a

given single mode to the member deformed configuration.

(44) Scale: Enter a scale factor for the 2D or 3D representation.

(45) 2D Plot: Pressing this button yields the 2D configuration (see Fig.

4.2(b)) related with the member (see (41)) and cross-section (see

(42)) selected. Moreover, the two radio buttons below allow the

choice between the in-plane (in-plane) or longitudinal

(warping) cross-section displacement fields. The undeformed

cross-section is represented in yellow, while the deformed in red.

(46) 3D Plot: Pressing the button creates, on a separate window, an

interactive interface (see Fig. 4.2(c)), where the whole member is

displayed and which contains several viewing tools. The deformed

configuration can also be plotted either with opaque surfaces

(Surface) or with a network of lines (Net).

(47) Plot options: Several options concerning the visualization of the

buckling load (or frequency) vs. length plot (see (48)):

Scale: Allows the choice between 3 possible scales for:

Logarithmic (Log), Bi-logarithmic (Log-log) or rectangular

(Rectangular). When changing the scale, the Change

limits checkbox must be unchecked.

Change Limits: Allows the user to change the top and bottom

limits of the plot. The checkbox Change limits must be on.

After writing the limits values, the user must press button

Update to update the plot.

Multiple plots: When this checkbox is checked, several curves,

corresponding to all the buckling (or vibration) modes calculated,

are plotted simultaneously. Otherwise, only the one related to the

first mode obtained is shown.

Show Markers: If this is checked, the points (i.e., solutions

obtained) used to trace the curve are represented by markers on

the plot.

(48) Buckling or frequency curves: The curves are traced by straight-

line segments linking the points corresponding to the solutions

obtained – there will be as many points as the number of lengths

provided. Several plot options are available (see (47)).

(49) Summary of results: Presents the main results concerning the

member selected (see (41)), namely: (i) the buckling load (or

frequency), and (ii) the modal participation percentages of the 3 most

important GBT modes.

(50) Modal participation diagram: The GBT modal participation

diagram (Pk-L) associated to the resulting buckling or vibration

modes. For more information on modal participation diagrams, see

GBT Theoretical Reference, Part 5.

4.2 Text Files

All the results are saved into three formatted text files, which can be opened

and used as input to most data processing applications. Those files, created

in the folder GBTUL\calc\Output_Files\, are the following:

(i) Matrices.txt – contains (i1) the displacement values (ui, vi and wi) 2 at

each cross-section node, for each deformation mode, and (i2) the

components of the GBT matrices (stiffness and mass).

(ii) Results.txt – includes (ii1) a list of the eigenvalues (buckling load

parameters or natural frequencies) associated with every member length

and eigenvector (buckling or vibration mode), as well as (ii2) the

corresponding modal participation factors and (ii3) the number of half-

waves they exhibit (only for the analytical solution).

(iii) Mafuncs.txt – contains the longitudinal amplitude functions2 (k(x))

and their derivatives (k,x(x)) associated with every deformation mode

included in the analysis, for all member lengths and buckling or vibration

modes determined. These functions are defined by their values at a

finite set of cross-sections along the member length.

2 See GBT Theoretical Reference.

Screen 4 – Results

Figure 4.1: GBTUL 1.0 – Overview of Screen 4

Case Selection -

see (41)

GBT modes – see

(43) and Fig. 4.2(a)

Buckling load

or frequency vs.

length curves

– see (48)

Cross-Section –

see (42)

Scale– see (44)

2D Plot – see (45)

and Fig. 4.2(b)

Plot options – see (47)

Modal participation

diagram – see (50)

3D Plot – see (46)

and Fig. 4.2(c)

Summary of

results – see (49)

Screen 4 – Results: Dialogue Box and Graphic Windows

(a) – see (43) (b) – see (45) (c) – see (46)

Figure 4.2: GBTUL 1.0 – Screen 4: (a) Modal selection dialogue box and (b) 2D and (c) 3D graphic representation window